| L(s) = 1 | + 16·4-s − 108·11-s + 192·16-s − 972·23-s + 1.82e3·25-s − 3.24e3·29-s + 892·37-s + 2.34e3·43-s − 1.72e3·44-s + 5.50e3·53-s + 2.04e3·64-s + 1.01e4·67-s − 1.87e4·71-s − 1.58e3·79-s − 1.55e4·92-s + 2.91e4·100-s − 6.23e4·107-s − 5.64e4·109-s + 3.82e4·113-s − 5.18e4·116-s − 5.06e4·121-s + 127-s + 131-s + 137-s + 139-s + 1.42e4·148-s + 149-s + ⋯ |
| L(s) = 1 | + 4-s − 0.892·11-s + 3/4·16-s − 1.83·23-s + 2.91·25-s − 3.85·29-s + 0.651·37-s + 1.26·43-s − 0.892·44-s + 1.96·53-s + 1/2·64-s + 2.25·67-s − 3.72·71-s − 0.254·79-s − 1.83·92-s + 2.91·100-s − 5.44·107-s − 4.74·109-s + 2.99·113-s − 3.85·116-s − 3.46·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 0.651·148-s + 4.50e−5·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.217340980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.217340980\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 1822 T^{2} + 1517859 T^{4} - 1822 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 54 T + 29723 T^{2} + 54 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 51652 T^{2} + 2274555846 T^{4} - 51652 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 144478 T^{2} + 12266392995 T^{4} - 144478 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 132334 T^{2} + 38286877971 T^{4} - 132334 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 486 T + 157931 T^{2} + 486 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 1620 T + 2066054 T^{2} + 1620 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2089198 T^{2} + 2719697478483 T^{4} - 2089198 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 446 T + 3214851 T^{2} - 446 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 9195268 T^{2} + 37043496050310 T^{4} - 9195268 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 1172 T + 3821766 T^{2} - 1172 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 14643502 T^{2} + 100836935539731 T^{4} - 14643502 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 2754 T + 12884483 T^{2} - 2754 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 21959950 T^{2} + 388143933008499 T^{4} - 21959950 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 19394686 T^{2} + 220056093143619 T^{4} - 19394686 p^{8} T^{6} + p^{16} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 5062 T + 24289995 T^{2} - 5062 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 9396 T + 71231078 T^{2} + 9396 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 24981118 T^{2} - 164639924559357 T^{4} - 24981118 p^{8} T^{6} + p^{16} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 794 T + 70499499 T^{2} + 794 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 153397060 T^{2} + 10369989980918982 T^{4} - 153397060 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 209766910 T^{2} + 18477749123078019 T^{4} - 209766910 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 281924740 T^{2} + 35525126061727494 T^{4} - 281924740 p^{8} T^{6} + p^{16} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.59422935798486340018437888828, −6.52261399559432525997395542633, −6.35760818632232266353172031271, −5.87584636097246163066757767760, −5.63356285209825316255541369620, −5.52205768244070662906492135783, −5.43087866380397271117764288696, −5.27434640911826518865529906378, −4.89906136807697180829591196361, −4.42184526181481728661969053155, −4.19220407093578355025768451577, −4.04392028461317290313097631270, −3.81781237986404836218285408034, −3.68769144337926929494380187550, −2.91996399180446479787533403947, −2.91752497365027447784724972894, −2.83967398720702717457000692203, −2.44564049374025048172006511601, −2.20185951989549439389420992902, −1.71730700285342166648513451371, −1.50999558459574949783856015422, −1.49096884732988222620748878222, −0.68683634125390101102052429953, −0.62307426048464318924639486627, −0.19834127206372104568255574462,
0.19834127206372104568255574462, 0.62307426048464318924639486627, 0.68683634125390101102052429953, 1.49096884732988222620748878222, 1.50999558459574949783856015422, 1.71730700285342166648513451371, 2.20185951989549439389420992902, 2.44564049374025048172006511601, 2.83967398720702717457000692203, 2.91752497365027447784724972894, 2.91996399180446479787533403947, 3.68769144337926929494380187550, 3.81781237986404836218285408034, 4.04392028461317290313097631270, 4.19220407093578355025768451577, 4.42184526181481728661969053155, 4.89906136807697180829591196361, 5.27434640911826518865529906378, 5.43087866380397271117764288696, 5.52205768244070662906492135783, 5.63356285209825316255541369620, 5.87584636097246163066757767760, 6.35760818632232266353172031271, 6.52261399559432525997395542633, 6.59422935798486340018437888828
